THE DOUBTER
returns to the subject in 1861, and seeks the reason why the
general equation of the fifth degree is solvable hi the manner in
which it is, thus taking a step beyond Abel who settled the
question of solvability 'by radicals'.
Much of Kronecker's work has a distinct arithmetical tinge,
either of rational arithmetic or of the broader arithmetic of
algebraic numbers. Indeed, if his mathematical activity had
any guiding clue, it may be said to have been his desire, perhaps
subconscious, to arithmetize all mathematics, from algebra to
analysis. 'God made the integers', he said, 'all the rest is the
work of man.* Kroneeker's demand that analysis be replaced by
finite arithmetic was the root of his disagreement with Weier-
strass. Universal arithmetization may be too narrow an ideal
for the luxuriance of modern mathematics, but at least it has
the merit of greater clarity than is to be found in some others.
Geometry never seriously attracted Kronecker. The period
of specialization was already well advanced when Kronecker
did most of his work, and it would probably have been impos-
sible for any man to have done the profoundly perfect sort of
work that Kronecker did as an algebraist and in Ms own
peculiar type of analysis and at the same time have accom-
plished anything of significance in other fields. Specialization is
frequently damned, but it has its virtues.
A distinguishing feature of many of Kronecker's technical
discoveries was the intimate way in which he wove together the
three strands of his greatest interests - the theory of numbers,
the theory of equations, and elliptic functions - into one beau-
tiful pattern in which unforeseen symmetries were revealed as
the design developed and many details were unexpectedly
imaged in others far away. Each of the tools with which he
worked seemed to have been designed by fate for the more
efficient functioning of the others. Not content to accept this
mysterious unity as a mere mystery, Kronecker sought and
found its underlying structure in Gauss9 theory of binary
quadratic forms, in which the main problem is to investigate
the solutions in integers of mdetenninate equations of tfce
second degree in two unknowns.
Kronecker's great work in the theory of algebraic numbers
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